L(s) = 1 | − 2·4-s + 7-s + 3·11-s − 13-s + 4·16-s − 3·17-s − 4·19-s − 9·23-s − 2·28-s + 6·29-s + 2·31-s + 37-s + 3·41-s − 2·43-s − 6·44-s − 6·47-s − 6·49-s + 2·52-s + 9·53-s + 12·59-s + 5·61-s − 8·64-s + 4·67-s + 6·68-s − 9·71-s − 14·73-s + 8·76-s + ⋯ |
L(s) = 1 | − 4-s + 0.377·7-s + 0.904·11-s − 0.277·13-s + 16-s − 0.727·17-s − 0.917·19-s − 1.87·23-s − 0.377·28-s + 1.11·29-s + 0.359·31-s + 0.164·37-s + 0.468·41-s − 0.304·43-s − 0.904·44-s − 0.875·47-s − 6/7·49-s + 0.277·52-s + 1.23·53-s + 1.56·59-s + 0.640·61-s − 64-s + 0.488·67-s + 0.727·68-s − 1.06·71-s − 1.63·73-s + 0.917·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.492412477320767612496192676444, −7.86809744691158498969719830425, −6.76645673034580029340057189178, −6.11005687269025714279511542679, −5.17476573972367257507257169598, −4.25960261379844898363305917120, −3.97212069424474633277892264236, −2.56854601966758836508468679405, −1.41149241933237558048400936281, 0,
1.41149241933237558048400936281, 2.56854601966758836508468679405, 3.97212069424474633277892264236, 4.25960261379844898363305917120, 5.17476573972367257507257169598, 6.11005687269025714279511542679, 6.76645673034580029340057189178, 7.86809744691158498969719830425, 8.492412477320767612496192676444